Topological Effects in Quantum Field Theories
Topology, the mathematical description of the robustness of form, appears in many branches of physics and provides strong constraints on physical systems. Topologically interesting field theories predict various hypothetical objects and mathematical constructs. Although many of the fascinating objects have not been seen in particle physics, they demonstrate possible phenomena that quantum field theory can support. A significant deformation of the Dirac equation is achieved when the constant mass term is replaced by topologically non-trivial profiles. Examples of interesting profiles are: kink in one dimension, vortex in two and magnetic monopole in three. Dirac equation in the presence of such topological entities possesses an isolated zero-energy solution. The existence of zero-energy solution does not require a speciﬁc form for the inhomogeneous mass proﬁle; all that matters is that it belongs to a non-trivial topological class. It has been shown that they give rise to fermion fractionalization and isolated Majorana fermion bound states depending on the absence or presence of superconductivity in the system. The difference between the two situations results in the presence or absence of a conserved fermion number.